Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-20T05:29:41.007Z Has data issue: false hasContentIssue false
  • English
  • Français

REFERENCE IN ARITHMETIC

Published online by Cambridge University Press:  14 January 2018

LAVINIA PICOLLO
LAVINIA PICOLLO*
Affiliation:
Ludwig-Maximilian University of Munich
*
*MUNICH CENTER FOR MATHEMATICAL PHILOSOPHY LMU MUNICH MUNICH, GERMANY E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Self-reference has played a prominent role in the development of metamathematics in the past century, starting with Gödel’s first incompleteness theorem. Given the nature of this and other results in the area, the informal understanding of self-reference in arithmetic has sufficed so far. Recently, however, it has been argued that for other related issues in metamathematics and philosophical logic a precise notion of self-reference and, more generally, reference is actually required. These notions have been so far elusive and are surrounded by an aura of scepticism that has kept most philosophers away. In this paper I suggest we shouldn’t give up all hope. First, I introduce the reader to these issues. Second, I discuss the conditions a good notion of reference in arithmetic must satisfy. Accordingly, I then introduce adequate notions of reference for the language of first-order arithmetic, which I show to be fruitful for addressing the aforementioned issues in metamathematics.

Keywords

03-XXreferencearithmeticself-referencediagonalization
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 3 , September 2018 , pp. 573 - 603
Copyright
Copyright © Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Boolos, G., Burgess, J. P., & Jeffrey, R. C. (2007). Computability and Logic (fifth edition). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Carnap, R. (1937). Logical Syntax of Language. London: Routledge.Google Scholar
Cook, R. T. (2006). There are non-circular paradoxes (but Yablo’s isn’t one of them!). The Monist, 89, 118–149.CrossRefGoogle Scholar
Cresswell, M. J. (1975). Hyperintensional logic. Studia Logica, 34, 25–38.CrossRefGoogle Scholar
Gödel, K. (1931). Über formal unentscheidebarre Sätze der Principia Mathematica und verwandler System I. Monathshefte für Mathematik und Physik, 38, 173–198.CrossRefGoogle Scholar
Goodman, N. (1961). About. Mind, 70, 1–24.CrossRefGoogle Scholar
Hájek, P. & Pudlák, P. (1993). Metamathematics of First-Order Arithmetic. Berlin: Springer.CrossRefGoogle Scholar
Halbach, V. (2009). Reducing compositional to disquotational truth. Review of Symbolic Logic, 2, 786–798.CrossRefGoogle Scholar
Halbach, V. (2016). The root of evil. A self-referential play in one act. In van Eijck, J., Iemhoff, R., and Joosten, J. J., editors. Liber Amicorum Alberti. A Tribute to Albert Visser. London: College Publications, pp. 155–163.Google Scholar
Halbach, V. & Visser, A. (2014a). Self-reference in arithmetic I. Review of Symbolic Logic, 7, 671–691.CrossRefGoogle Scholar
Halbach, V. & Visser, A. (2014b). Self-reference in arithmetic II. Review of Symbolic Logic, 7, 692–712.CrossRefGoogle Scholar
Heck, R Jr.. (2007). Self-reference and the languages of arithmetic. Philosophia Mathematica, III, 1–29.Google Scholar
Henkin, L. (1952). A problem concerning provability. The Journal of Symbolic Logic, 17, 160.Google Scholar
Henkin, L. (1954). Review of G. Kreisel: On a problem of Henkin’s. The Journal of Symbolic Logic, 19, 219–220.CrossRefGoogle Scholar
Horwich, P. (1998). Truth (second edition). New York: Blackwell.CrossRefGoogle Scholar
Kaye, R. (1991). Models of Peano Arithmetic. Oxford: Clarendon Press.Google Scholar
Kleene, S. (1938). On notation for ordinal numbers. The Journal of Symbolic Logic, 3, 150–155.CrossRefGoogle Scholar
Kreisel, G. (1953). On a problem of Henkin’s. Indagationes Mathematicae, 15, 405–406.CrossRefGoogle Scholar
Leitgeb, H. (2002). What is a self-referential sentence? Critical remarks on the alleged (non)-circularity of Yablo’s Paradox. Logique et Analyse, 177–178, 3–14.Google Scholar
Leitgeb, H. (2005). What truth depends on. Journal of Philosphical Logic, 34, 155–192.CrossRefGoogle Scholar
Löb, M. H. (1955). Solution of a problem of Leon Henkin. The Journal of Symbolic Logic, 20, 115–118.CrossRefGoogle Scholar
Milne, P. (2007). On Gödel sentences and what they say. Philosophia Mathematica, III(15), 193–226.CrossRefGoogle Scholar
Montague, R. (1962). Theories incomparable with respect to relative interpretability. The Journal of Symbolic Logic, 27, 195–211.CrossRefGoogle Scholar
Priest, G. (1997). Yablo’s paradox. Analysis, 57, 236–242.CrossRefGoogle Scholar
Putnam, H. (1958). Formalization of the concept of about. Philosophy of Science, 25, 125–130.Google Scholar
Ryle, G. (1933). Imaginary objects. Proceedings of the Aristotelian Society, 12(Suppl.), 18–43.CrossRefGoogle Scholar
Smoryński, C. (1981). Fifty years of self-reference in arithmetic. Notre Dame Journal of Formal Logic, 22(4), 357–374.CrossRefGoogle Scholar
Smoryński, C. (1991). The development of self-reference: Löb’s theorem. In Drucker, T., editor. Perspectives on the History of Mathematical Logic. Boston: Birkhäuser, pp. 110–133.Google Scholar
Sorensen, R. A. (1998). Yablo’s paradox and kindred infinite liars. Mind, 107, 137–155.CrossRefGoogle Scholar
Tarski, A. (1935). Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica Commentarii Societatis Philosophicae Polonorum, 1, 261–405, reprinted as The Concept of Truth in Formalized Languages in Logic, Semantics and Metamathematics, pp. 152–278.Google Scholar
Tarski, A. (1944). The semantic conception of truth: And the foundations of semantics. Philosophy and Phenomenological Research, 4, 341–376.CrossRefGoogle Scholar
Urbaniak, R. (2009). Leitgeb, “About,” Yablo. Logique et Analyse, 207, 239–254.Google Scholar
Visser, A. (1989). Semantics and the liar paradox. In Gabbay, D. M. and Günthner, F., editors. Handbook of Philosophical Logic, Vol. 4. Dordrecht: Reidel, pp. 617–706.CrossRefGoogle Scholar
Yablo, S. (1985). Truth and reflexion. Journal of Philosphical Logic, 14, 297–349.Google Scholar
Yablo, S. (1993). Paradox without self-reference. Analysis, 53, 251–252.CrossRefGoogle Scholar